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Functors in Differential Geometry

De Waal Martijn - Smeenk Sebastiaan - Sitbon Pascal - Zacharopoulos Georgios

February 4, 2019

Abstract

After presenting the notion of a category and providing some examples we focus towards the

cotangent bundle of a manifold. This space plays a very important role in symplectic geometry.

Indeed we will show that it is possible to give a symplectic structure on the cotangent bundle

in a canonical way. We present a functor between the category of smooth manifolds equipped

with diffeomorphisms (or local diffeomorphisms) and the category of symplectic manifolds

equipped with symplectomorphisms for which the object assignment consists of associating to

a space its cotangent bundle.

1

Contents

1 Introduction

3

2 A brief history of Vector Bundles and Symplectic Manifolds

4

3 Categories and Functors

3.1 Definitions . . . . . . . . . . . . . .

3.2 Examples in Algebraic Topology .

3.2.1 Linearization . . . . . . . .

3.2.2 Fundamental Group . . . .

3.3 Examples in Differential Geometry

3.3.1 Smooth Maps . . . . . . . .

3.3.2 Tangent Spaces . . . . . . .

3.3.3 de Rham Cohomology . . .

3.3.4 The Tangent Bundle . . . .

3.3.5 The Cotangent Bundle . . .

3.3.6 Smooth Functors . . . . . .

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4 The Symplectic Cotangent Functor

4.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 The Cotangent Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

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1

Introduction

Category theory is the study of mathematical structures and relations between them. It provides

a common language to almost all areas of mathematics, its power as a language can be compared

to the importance of basic set theory, but in a more general framework. Indeed we can for instance

consider the category of manifolds while the class of all manifolds is not a set. It is also interesting

to look at the relationships between categories which we call functors. Moreover it happens quite

often in mathematics that one talks about natural constructions or transformations. Category

theory allows to define what natural means via the notion of natural transformations which can

be thought as the relationships between functors of two categories. As a consequence it allows to

actually prove that a construction is natural which is very handy for mathematicians. Indeed the

word natural comes up quite often in mathematical constructions. A natural construction is usually

understood as a new construction that emerges directly (naturally) from previous operations and

constructions, and for which we feel like there is just one way to do the actual construction (e.g. if V

is a vector space of dimension n there is a natural isomorphism between V and V ∗∗ ). More precisely,

in category theory, one talks about objects and morphisms (Figure 1), which are maps between

these objects, both satisfying the defining properties of the category. A well-known example of a

category is the category of groups, together with the definition of a group homomorphism. After

defining these notions, we present some important and familiar examples of categories and functors

in Differential Geometry and Algebraic Topology. The most relevant functor for our project is the

cotangent functor. We will see how the contangent bundle can be viewed as a functor between

the category of Smooth Manifolds (equipped with diffeomorphisms) and Symplectic Manifolds

(equipped with symplectomorphisms) by proving the following theorem (in section 4.2):

Theorem 4.7: The canonical symplectic form on the cotangent bundle is invariant under

˜ are smooth manifolds and F : Q → Q

˜

diffeomorphisms in the following sense : Suppose Q and Q

˜ → T ∗ Q be the map described in section 3.3.5. T ∗ F is a

a diffeomorphism. Let T ∗ F : T ∗ Q

˜ are endowed with their canonical symplectic forms.

symplectomorphism when both T ∗ Q and T ∗ Q

Figure 1: Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ◦ f .

(Identities would appear as three arrows, from the letters X, Y , and Z to themselves)

Y. Eliashberg, a symplectic geometer, asked a natural question following the construction of

the cotangent functor, precisely is it injective in the following sense:

˜ are such that T ∗ Q and T ∗ Q

˜ are symplectomorphic then

Question: If two manifolds Q and Q

˜

are Q and Q diffeomorphic?

This is a hard question that is still an open problem in differential geometry. F. Ziltener and Y.

Karshon [4] answer this question in a particular case. Namely, if G is a compact and connected

Lie group, they classify the Hamiltonian Lie group actions up to equivariant symplectomorphisms,

by providing an injective functor between the category of Hamiltonian Lie group actions and the

category of symplectic representation of closed subgroup of G, an Hamiltonian Lie group action

being canonically symplectomorphic to the cotangent bundle symplectic quotient, this provides an

answer to the question in this precise case. Another case for which injectivity holds, is the case of

exotic spheres of odd dimension ≥ 5. The first breakthrough was made by Abouzaid[7] who showed

that for 4k + 1 dimensional homotopy spheres that do not bound a parallelisable manifold, the

cotangent bundle of these spheres is not diffeomorphic to T ∗ S 4k+1 . Later this result was enhanced

by Ekholm et al [8] who proved that up to orientation preserving diffeomorphism all exotic spheres

have distinct cotangent bundles.

3

2

A brief history of Vector Bundles and Symplectic Manifolds

The vast development of physics in the early years of the 20th century had a huge impact in

mathematics. "The astonishing fact that gravitation is just a manifestation of the curvature

of spacetime made a deep impression on mathematicians like Elie Cartan and Hermann Weyl.

Especially Weyl, who was a student of Hilbert at that time". [5]

His initial work was on analysis and spectral theory, but he made a lot of contributions to differential

geometry too, motivated by the work of the Italian geometers, like Ricci and Levi-Civita, who had

studied deeply the aspects of curvature and tensor calclus on Riemannian manifolds. A serious

obstruction in doing physics or mathematics on manifolds is that the result have to be checked in

all the coordinates systems of the manifold, if the manifold itself do not have an intrinsic defined

coordinate system. A method to overcome this obstruction is by making computations with tensors

and their derivatives. That coordinate-invariant differential calculus of tensors can be realized via

the existence of a connection or, equivalently, an isomorphism between the tangent spaces at a

neighbourhood of a point. This isomorphism is now called the Parallel transport.

Parallel transport is a way of moving tangent vectors along curves without changing their angle

and their length. By define parallel transport one can define derivatives of vector and tensor fields.

But Levi-Civita, who has already used this isomorhism, only worked for Riemannian manifolds

that were embedded in an Euclidean space. It was Weyl who defined the parallel transport on any

Riemannian manifolds not necessarily embedded in an Euclidean space and not only that, he also

observed that in order to define the parallel transport it was necessary that the manifold should

have a metric. "Therefore he defined the concept of parallel transport in a more axiomatic way.

He created the notion of an affine connection on a Riemannian manifold". [5]

Weyl then tried to generalize the concept of parallel transport by scaling the length of the vectors

and by considering the possible scales at the various points as a bundle on the manifold. He

introduced connections on this bundle and with the use of differential forms defined on the manifold

he expressed the curvature of this bundle. This vector bundle is called the scale bundle, as it is

defined in [5], and it was one of the first appearances of vector bundles in mathematics, apart from

the usual tangent bundle.

"The origin of Symplectic Geometry lies in Classical Mechanics", [6]. The equations of motion in Classical Mechanics arise as solutions of variational problems. A characteristic example

is Fermat’s principle of least time which states that light moves from one point to another by a

path which takes the shortest amount of time. Similarly, all systems that possess kinetic but not

potential energy move along geodesics, which are paths minimizing length and energy.

If a system has kinetic and potential energy then there is a quantity which minimize the mean

value of kinetic and potential energy. This quantity is called the action. The paths that minimize

some action functional are solutions of a system of n second order differential equations called

the Euler-Lagrange equations of the variational problem. The study of Euler-Lagrange equations

of one-dimensional variation and the Hamiltonian formalism leads to the notion of a canonical

transformation which preserves Hamilton’s form of the equations of motion. Based on this observation one can define a manifold equipped with an atlas whose transition maps are the canonical

transformations. Such a manifold is called symplectic. This structure give rise to a differential

form which is called the symplectic form.

A manifold equipped with a symplectic form is called a symplectic manifold. The symplectic form

is closed and non-degenerate and from the skew-symmetry of differential forms it follows that

manifolds carrying such forms have to be even dimensional. The canonical transformations are

called symplectomorphisms and they are precisely the maps that preserve the symplectic form of a

manifold. "The mathematical formulation of classical mechanics gave rise to this beautiful subject

of mathematics that is now called Symplectic Geometry and Topology", [6]. which is the subject

of mathematics that studies manifolds equipped with symplectic structure. An astonishing fact

is that the general theory of vector bundles had a huge impact on the development of symplectic

geometry. More precisely the contagent bundle of every smooth manifold has a natural symplectic

structure which is one of the main topics of this report, (section 4.2). Also, symplectic Geometry

has many applications apart from physics, also in other areas of mathematics like low-dimensional

topology, homological algebra, representation theory, algebraic and complex geometry and category

theory. [6]

4

3

Categories and Functors

3.1

Definitions

In order to read this report a basic understanding of category theory is needed. Here we introduce

three important concepts: categories, functors and natural transformations.

Definition 3.1.1. A category C is a collection of objects Ob(C), a collection of morphisms Hom(C)

between those objects and a composition operator ◦ defined on the morphisms. We denote the morphisms between objects X, Y ∈ Ob(C) by HomC (X, Y ). Furthermore, the following properties

hold:

1. Composition of morphisms is associative: For objects W, X, Y, Z ∈ Ob(C) and morphisms

f : Y → Z, g : X → Y and h : W → X (so f ∈ HomC (Y, Z) etc. . . ) we have

h ◦ (g ◦ f ) = (h ◦ g) ◦ f.

2. For every object there exists an identity: For every X ∈ Ob(C) there exists a function

1X ∈ HomX, X (C) such that 1X ◦ f = f and g ◦ 1X = g for any morphisms f, g with

codomain X and domain X respectively.

4

The notation Ob(C) and Hom(C) will be used in this section in order to define category

theoretical concepts formally, but in the remainder of this report the notation will be dropped and

the categories with their respective objects, morphisms and compositions will be clear.

Example 3.1.2. A very simple example is the category of sets and their maps: Sets where the

objects are sets, maps between sets are morphisms and the standard composition is the composition

operator. Of course, one can restrict the sets to topological spaces, metric spaces, groups and so

forth, while restricting the maps to continuous maps or homomorphisms. In those cases there is

still an identity map and composition is still associative.

4

Next, we need the concept of a map between categories.

Definition 3.1.3. Let C, D be categories. A functor F : C → D is a map that works on objects

and morphisms by F : Ob(C) → Ob(D) and F : Hom(C) → Hom(D) such that the category

structure is preserved:

1. If f ∈ HomC (X, Y ), g ∈ HomC (Y, Z) then F (g ◦ f ) = F (g) ◦ F (f ). In this case F is called

a covariant functor.

2. For every X ∈ Ob(C) we have F (1X ) = 1F (X) .

4

More specifically, the above definition is that of a covariant functor. This means that the

order of composition is preserved. This is in contrast with a contravariant functor for which every

morphism f : X → Y is sent to F (f ) : F (Y ) → F (X). As a result F (g ◦ f ) = F (f ) ◦ F (g).

Lastly, one can map functors to functors by means of a natural transformation.

Definition 3.1.4. Given two functors F, G : C → D a natural transformation η : F → G is a

family of morphisms {ηX : F (X) → G(X)}X∈Ob(C) in Hom(D) such that the following diagram

is commutative:

F (X)

F (f )

ηX

G(X)

F (Y )

ηY

G(f )

5

G(Y )

If the functors F, G are contravariant F (f ) : F (Y ) → F (X) and G(f ) : G(Y ) → G(X) and

therefore the horizontal arrows in the above diagram should be flipped.

4

Example 3.1.5. The definition of a functor and a natural transformation allow us to consider the

collection of functors {F : C → D} and their natural transformations as a category. The functors

are the objects, and the natural transformations are the morphisms. Such a category is called a

functor category.

4

3.2

3.2.1

Examples in Algebraic Topology

Linearization

Let’s consider the two following categories:

• Set The objects are sets and where the morphisms are set maps.

• Ab The objects are Abelian groups and where the morphisms are group homomorphisms.

Consider a set X and an Abelian group A, then the A-linearization of X, denoted as A[X], is

defined as :

A[X] = {f : X → A|card(f −1 (A − {0})) < ∞}.

The A-linearization A[X] inherits the Abelian group structure from A, and it is useful to see

elements of A[X] as a formal finite linear combination with coefficients in A. If f ∈ A[X], it is

non-zero on a finite set of points of X, say {x1 , . . . , xn }. Denoting f (x1 ), . . . , f (xn ) by a1 , . . . , an

we can represent f as

n

X

f=

ai · xi .

i=1

From a set map γ : X → Y we get an induced map

A[X] → A[Y ]

γ∗ :

f → γ∗ (f )

where

γ∗ (f ) =

n

X

ai · γ(xi ).

i=1

Notice that, for any f ∈ A[X], we have (IdX )∗ (f ) = f . Also it is clear that (γ1 ◦γ2 )∗ = (γ1 )∗ ◦(γ2 )∗ .

Hence the rule F defined below is a covariant functor.

Set→ Ab

Fob :

X → A[X]

Fhom :

3.2.2

Set Maps→ Group Homomorphisms

γ → γ∗

Fundamental Group

Consider the two following categories:

• Top∗ where objects are pointed topological spaces and morphisms are pointed continuous

maps, and

• Grp where objects are groups and morphisms are group homomorphisms.

A continuous map f : (X, p) → (Y, f (p)) between pointed topological spaces induces a group

homomorphism f∗ : π1 (X, p) → π1 (Y, f (p)) defined as:

π1 (X, p) → π1 (Y, f (p))

f∗ :

[α] → [f α]

6

It’s direct to prove that that f∗ is a well-defined homomorphism and that (Id(X,p) )∗ = Idπ1 (X,p)

and that (f ◦ g)∗ = f∗ ◦ g∗ . Hence the rule F defined below is a covariant functor.

Top∗ → Grp

Fob :

(X, p) → π1 (X, p)

Fhom :

3.3

3.3.1

Pointed Continuous Maps → Group Homomorphisms

(f : (X, p) → (Y, f (p)) → f∗ : π1 (X, p) → π1 (Y, f (p))

Examples in Differential Geometry

Smooth Maps

Let’s consider two new categories:

• ManSmooth : The category where the objects are smooth manifolds and the morphisms are

smooth maps.

• VecR : The category where the objects are real vector spaces and the morphisms are linear

maps.

For a manifold M , we denote by C ∞ (M ) the set of smooth maps from M to R. The set C ∞ (M )

has a R-vector space structure by pointwise addition and scalar multiplication. Let’s now consider

two smooth manifolds M, N and a smooth map f : M → N . Then f induces a linear map f∗

defined as

∞

C (N ) → C ∞ (M )

f∗ :

g →g◦f

From this definition it is clear that (IdM )∗ = IdC ∞ (M ) and also that (f ◦ g)∗ = g∗ ◦ f∗ . This defines

the rule F below, which is a contravariant functor:

ManSmooth → VecR

Fob :

M → C ∞ (M )

Smooth Maps → Linear Maps

Fhom :

f → f∗

3.3.2

Tangent Spaces

In this example we consider the category VecR and a new one:

• ManSmooth∗ : The objects are pointed smooth manifolds and the morphisms are pointed

smooth maps.

A smooth map f : (M, p) → (N, f (p)) induces a linear map called the differential of f at p,

denoted by dfp : Tp M → Tf (p) N . We have that d(IdM )p = IdTp M . Furthermore the composition

rule ensures that d(g ◦ f )p = dgf (p) ◦ dfp . Hence the rule F defined below is a covariant functor.

ManSmooth∗ → VecR

Fob :

(M, p) → Tp M

Fhom :

3.3.3

Pointed Smooth Maps → Linear Maps

(f, p) → dfp

de Rham Cohomology

Let’s now go back to two previous categories, namely ManSmooth and Grp. Recall that we can

define the pull-back for any smooth map f : M → N between two manifolds M and N denoted by

f ∗ . If ω ∈ Ωn (N ) then we define f ∗ ω ∈ Ωn (M ) as follows, for v1 , . . . , vn ∈ Tp M :

(f ∗ ω)p (v1 , . . . , vn ) = ωf (p) (dfp (v1 ), . . . , dfp (vn ))

7

The pull-back of any smooth map f : M → N between two manifolds M and N , denoted by f ∗

induces an algebra homomorphism

HdR (N ) → HdR (M )

f∗ :

[ω] → [f ∗ ω]

Moreover, (IdM )∗ = IdHdR (M ) , and since the pullback of a composition is contravariant the induced

homomorphism inherits from it. Hence the rule F defined below is a contravariant functor.

ManSmooth → Grp

Fob :

M → HdR (M )

Smooth Maps → Group Homomorphisms

Fhom :

f → f∗

3.3.4

The Tangent Bundle

The tangent bundle is defined as

TM =

G

Tp M.

p∈M

It has the structure of smooth manifold with dimension 2n if M has dimension n. If f is a smooth

map between two manifolds M and N then the total derivative of f , denoted by df , is a map

between tangent bundles.

TM → TN

df :

(p, v) → (f (p), dfp (v))

The composition rule ensure that the rule F defined below is a covariant functor.

ManSmooth → ManSmooth

Fob :

M → TM

Smooth Maps → Smooth maps

Fhom :

f → df

3.3.5

The Cotangent Bundle

The cotangent bundle of a smooth manifold M is defined as follows

G

T ∗M =

Tp∗ M.

p∈M

It is the disjoint union of all dual spaces at each point of the manifold. There is a nice way to

interpret the cotangent bundle in physics for instance. Indeed, as explained in [9], "If the manifold

M represents the set of possible positions in a dynamical system, then the cotangent bundle T ∗ M

can be thought as representing the set of possible positions and momenta. For example, this is a

way to describe the phase space of a pendulum. The state of the pendulum is determined by its

position (an angle) and its momentum. The entire state space looks like a cylinder". The cylinder

S 1 × R is the cotangent bundle of the circle (it is the trivial line bundle over S 1 ).

For any manifold M of dimension n the cotangent bundle of M has a structure of smooth manifold

of dimension 2n. Later in this report we will see that the cotangent bundle has a natural symplectic

manifold structure, hence we will be able to take a look at the cotangent bundle within the

Symplectic Manifold category. It is also a smooth vector bundle over M of rank n with its standard

projection map π : T ∗ M → M and the natural vector space on each fiber. Let p ∈ M , consider a

chart (U, φ) with p ∈ U and φ = (x1 , . . . , xn ). We define the following local frame

U → T ∗ M |U

i

dx :

p → dxip ∈ Tp∗ M

∂

|p ) = δij (here we make the identification Tp R and R). The family dx1 |U , . . . , dxn |U

where dxip ( ∂x

j

is a local frame over U (the sections being smooth). They are called coordinate covector fields.

Since being trivializable over an open set is equivalent to admitting a local frame over U , the

sections of the frame being smooth proves that the cotangent bundle is a smooth vector bundle of

rank n.

From now on we will consider the following categories:

8

• VB : The objects are smooth vector bundles and the morphisms are smooth bundle homomorphisms,

• ManDiff : The objects are smooth manifolds and the morphisms are diffeomorphisms.

Proposition 3.1: Suppose F : M → N a diffeomorphism, and let T ∗ F : T ∗ N → T ∗ M be the

map whose restriction to each Tq∗ N is dFF∗ −1 (q) , then T ∗ F is a smooth bundle homomorphism.

Proof: If we denote by πM and πN the standard projections of T ∗ M and T ∗ N , we have to

show that the following diagram commutes.

T ∗N

T ∗F

T ∗M

πN

πM

N

F −1

M

If we denote by (V, ψ = (y 1 , . . . , y n )) a chart at q ∈ N , the coordinate covector fields dy 1 , . . . , dy n

allow us to write an element of ω ∈ Tq∗ N as

ω = ξ i dy i |q .

Hence

F −1 ◦ πN (ω) = F −1 (q),

πM ◦ T ∗ F (ω) = πM ◦ dFF∗ −1 (q) (ω) = F −1 (q)

since dFF∗ −1 (q) (ω) ∈ TF∗ −1 (q) M . Also dFF∗ −1 (q) : Tq∗ N → TF∗ −1 (q) M is a linear map, so T ∗ F is

linear on each fibers. Lastly, F is assumed to be a diffeomorphism so F −1 is smooth. We conclude

that T ∗ F is a smooth bundle homomorphism.

Proposition 3.2: The assignment M 7→ T ∗ M , F 7→ T ∗ F is a contravariant functor from

ManDiff to VB. It is called the cotangent functor.

Proof: Let us denote such a rule by F. We just saw that if F ∈ HomManDiff (M, N ), then

dF ∗ ∈ HomVB (F(N ), F(M )). We need to show that

∀M ∈ Ob(ManDiff ), F(IdM ) = IdF (M )

∀(f, g) ∈ HomManDiff (M, N ) × HomManDiff (P, M ), F(f ◦ g) = F(g) ◦ F(f ).

Let M be a smooth manifold. IdM is indeed a diffeomorphism as it is smooth and involutive.

Moreover, for any element p ∈ M , dIdM = IdT M . Hence, dId∗M = Id∗T M . Or, for any ω ∈ Tp∗ M and

v ∈ Tp M , dId∗M (ω)(v) = ω(dIdp (v)) = ω(v), i.e. dId∗M = IdT ∗ M . This proves the first statement.

The second statements follows from the fact that the differential is covariant and the pull-back is

contravariant. For two maps G : M → N and F : N → P . For ω ∈ Tp∗ P and v ∈ Tp P , we have

T ∗ (F ◦ G)(ω)(v) = ω(d(F ◦ G)∗(F ◦G)−1 (p) (v))

= ω((dFF −1 (p) ◦ dGG−1 ◦F −1 (p) )∗ (v))

= ω(dG∗G−1 ◦F −1 (p) ◦ dFF∗ −1 (p) (v))

(1)

= T ∗ G ◦ T ∗ F (ω)(v),

hence T ∗ (F ◦ G) = T ∗ G ◦ T ∗ F . We finally conclude that T ∗ is a contravariant functor. The

following section will introduce symplectic manifolds and present the category of symplectic manifolds. The aim will be to present a functor from the category of smooth manifolds equipped with

diffeomorphisms to the category of symplectic manifolds equipped with symplectomorphisms, for

which a smooth manifold M is mapped to T ∗ M .

9

3.3.6

Smooth Functors

It is possible to construct vector bundles out of old ones. This is because one can construct new

vector space out of old ones via classic operations on vector spaces.

Example 3.3 The Whitney Sum:

If V and W are vector space then V ⊕ W is a new vector space. Then, if E → M , F → M

are two vector bundles, we can define (E ⊕ F )x = Ex ⊕ Fx , yielding a collection of vector spaces

out of which it is possible to construct a vector bundle. If {gβα }, (resp. {hα

β }) are the transition

functions of E (resp. F ), then we

can construct

a

vector

bundle,

for

which

the

fibers are Ex ⊕ Fx

gβα 0

and the transitions functions are

. This can be generalized to many other operations

0 hα

β

V

V

k

k

: E∗, E ⊗ F ,

E, E ∗ . . . . As presented in [2], ”The framework which justifies why linear

algebraic constructions carry through to vector bundles uses the language of categories.”. The

relevant category in this framework is the following:

• V, the category whose objects are finite dimensional real or complex vector spaces and whose

morphisms are linear isomorphisms.

We define in this precise case what a binary functor is:

Definition 3.4 A binary functor is a rule F : V × V → V such that:

• To each pair of vector spaces V, W it assigns a vector space F(V, W )

• To each pair of isomorphisms f, g it assigns an isomorphism F(f, g).

And also such that it is functorial in each of its variable :

• F(Id, Id) = Id

• F(f ◦ f 0 , g ◦ g 0 ) = F(f, g) ◦ F(f 0 , g 0 ).

Definition 3.5 A smooth (binary) functor is a functor F such that F(f ) (resp.F(f, g)) depends

smoothly on f and g.

As explained in [2], ’An intuitive and less intrinsic way to think of smooth functors is that bases

for V and W give rise to a basis for T (V, W ) and smooth changes on the bases of V and W causes

the basis of T (V, W ) to change smoothly.’ And this relates to our construction in the Whitney

E

F

F

sum. Indeed, local frames {sE

1 , . . . , sk } and {s1 , . . . , sl } on U give rise to the local frame

E

F

F

{(sE

1 , 0), . . . , (sk , 0), (0, s1 ), . . . , (0, sl )}.

α,E

α,F

α,F

β,E

The change of basis between {sα,E

}) and {sβ,E

1 , . . . , sk } (resp. {s1 , . . . , sl

1 , . . . , sk } (resp.

β,F

β,F

{s1 , . . . , sl }) is given by the smooth transition function gβα (resp. hα

β ). And change of basis on

the new frame is given by

α

gβ 0

0 hα

β

which is smooth because gβα and hα

β are smooth. Note that we can extend a binary functor to a

k-functor (defined similarly with k variables instead of two), hence generalize this point of view to

many operations on vector spaces.

10

4

The Symplectic Cotangent Functor

4.1

Symplectic Manifolds

In this section we give the basic definitions from symplectic geometry.

Definition 4.1 Let M 2d be a differentiable manifold of even dimension 2d. A symplectic

structure on M is a 2-form ω that is

• closed i.e. dω = 0,

• non-degenerate: for v, w ∈ Tx M , if ω(v, w) = 0 for all w ∈ Tx M , then v = 0.

A pair (M 2d , ω) is called a symplectic manifold.

Definition 4.2: Let (M1 , ω1 ) and (M2 , ω2 ) be two symplectic manifolds. A differentiable map

f : M1 → M2 is called symplectic if the pullback of ω2 under f is equal to the symplectic structure

on M1 :

f ∗ ω2 = ω1 .

A symplectic diffeomorphism is called a symplectomorphism.

Example 4.4: Let R2n be the Euclidean space of dimension 2n and let (x1 , . . . , xn , y 1 , . . . , y n )

be the standard coordinates. We define the 2-form

ω=

n

X

dxi ∧ dy i .

i

This form is symplectic, indeed

dω = d

n

X

dxi ∧ dy i =

n

X

d(dxi ∧ dy i ) = 0

i

i

and non-degenerate because its value at each point of R2n is the symplectic 2-tensor which is

non-degenerate.

Example 4.5: Another example of a symplectic manifold is the sphere S 2 . Let ω be a smooth

non-vanishing 2-form. Then ω is closed because the differential of a 2-form is a 3-form which,

defined on a 2-dimensional object, is zero and it’s non-degenerate because in two dimensions every

non-vanishing form is non-degenerate. This argument can be generalized and we can show that

actually every orientable smooth manifold of dimension two admits a symplectic structure.

Lemma 4.6: The collection Symp for which objects are symplectic manifolds and morphisms

are symplectomorphisms is a category.

Proof of Lemma 4.6 We can indeed compose morphisms because if f : (M1 , ω1 ) → (M2 , ω2 ) and

f : (M2 , ω2 ) → (M3 , ω3 ) are symplectomorphisms then h = g ◦ f : (M1 , ω1 ) → (M3 , ω3 ) since (g ◦

f )∗ ω3 = f ∗ ◦ g ∗ ω3 = f ∗ ω2 = ω1 . Composition is associative as classical composition is associative.

Notice that the identity map between two symplectic manifolds is a symplectomorphism because

the identity between two manifolds induces the identity on forms hence, if (M, ω) is a symplectic

manifold, then Id∗M ω = ω. This achieves to prove that Symp is a category.

4.2

The Cotangent Functor

The aim of this section is to present the category of symplectic manifolds endowed with symplectomorphisms and to show the following invariance result.

Theorem 4.7: The canonical symplectic form on the cotangent bundle is invariant under

˜ are smooth manifolds and F : Q → Q

˜

diffeomorphisms in the following sense: Suppose Q and Q

∗

∗˜

∗

∗

a diffeomorphism. Let T F : T Q → T Q be the map described in section 3.3.5. T F is a

˜ are endowed with their canonical symplectic forms.

symplectomorphism when both T ∗ Q, T ∗ Q

This result yields the following corollary:

11

Corollary 4.8: The assignment M 7→ T ∗ M , F 7→ T ∗ F is a contravariant functor from

ManDiff to Symp.

There is a natural 1-form λ on the total space of T ∗ M called the tautological 1-form that we

can obtain from the standard projection, for this purpose we follow the construction made by John

M. Lee in [1]. A point in T ∗ M can be represented as a covector φ ∈ Tq∗ M for some q ∈ M that we

denote by (q, φ). Recall that the standard projection is smooth and is represented by π(q, φ) = q.

The pointwise differential of π yields a linear map

dπ(q,φ) : T(q,φ)) (T ∗ M ) → Tq M

As explained in [3], in physics, ”M corresponds to configuration space and T ∗ M to phase space.

The point q corresponds to the (generalized) position of a particle and the covector p ∈ Tq∗ M to

its (generalized) momentum.” The pullback of this map yields a linear map

∗

∗

: Tq∗ M → T(q,φ)

(T ∗ M ).

dπ(q,φ)

Hence we can define the following 1-form λ ∈ Ω1 (T ∗ M )

∗

∗

λ(q,φ) = dπ(q,φ)

φ ∈ T(q,φ)

(T ∗ M ).

Proposition 4.9: Let M be a smooth manifold. The tautological 1-form is smooth, and ω =

−dλ is a symplectic form on the total space of T ∗ M .

To prove this, we need the following lemma:

Lemma 4.10: Let π : E → M be a smooth vector bundle of rank k. L et (U, φ) be a smooth

chart on M with coordinate (xi ), and suppose there exists a smooth local frame (σi ) for E over U .

Define φ˜ : π −1 (U ) → φ(U ) × Rk by

˜ i σi (p)) = (x1 (p), . . . , xn (p), v 1 , . . . , v k ).

φ(v

˜ is a smooth coordinate chart over E.

Then (π −1 (U ), φ)

Proof of Lemma 4.10: Every smooth local frame for a smooth vector bundle is associated with

a smooth local trivialization. Let Φ denote the smooth local trivialization over U that corresponds

to the local frame (σi ), it is defined by

Φ(v i σi (p)) = (p, (v 1 , . . . , v n )).

Then φ˜ = (φ × IdRk ) ◦ Φ, this proves that φ˜ is a smooth map.

Proof of Proposition 4.9: Let (U, φ = (xi )) be a chart. We saw that (dxi ) forms a smooth local

frame for T ∗ M over U . Now, Lemma 4.10 ensures that the map from π −1 (U ) to R2n given by

ξi dxi |p 7→ (x1 (p), . . . , xn (p), ξ1 , . . . , ξn )

is a smooth coordinate chart for T ∗ M . We call (xi , ξi ) the natural coordinates for T ∗ M associated

with (xi ). As shown in [1], λ has the following natural coordinates representation

∗

λ(x,ξ) = dπ(x,ξ)

(ξi dxi ) = ξi dxi .

Hence, λ is smooth because it is linear in its components. Furthermore, ω is exact and therefore

closed. We need to verify that it is non-degenerate. We have

ω = −dλ =

n

X

dxi ∧ dξi

i=1

which, under identification, corresponds to the standard symplectic 2-form in R2n .

Note that there is a reciprocity theorem, called the Darboux Theorem that won’t be proved

here, a proof can be found in [1]. The result is the following:

12

Theorem 4.11: Let (M, ω) be a 2n-dimensional symplectic manifold. For any p ∈ M there

are smooth coordinates (x1 , . . . , xn , y 1 , . . . , y n ) centered at p in which ω has the coordinate representation:

n

X

ω=

dxi ∧ dy i .

i=1

The symplectic form ω defined in proposition 4.9 is called the canonical symplectic form on

T ∗ M . Now we have the tools to prove Theorem 4.7, since we have a symplectic structure on the

cotangent bundle.

˜ are smooth manifolds and F : Q → Q

˜ a diffeomorProof of Theorem 4.7: Suppose Q and Q

˜

∗

∗˜

∗

phism. Let T F : T Q → T Q be the morphism described in section 3.3.5, and denote by ω Q , ω Q

∗

∗˜

∗

the canonical symplectic forms on T Q and T Q, respectively. Denote Φ = T F , we need to show

˜

˜

that Φ∗ ω Q = ω Q . Notice that it suffices to show that Φ∗ λQ = λQ , so we compute Φ∗ λQ q˜,φ˜:

Φ∗ λ Q

˜

q˜,φ

= λQ

˜

˜ dΦq˜,φ

Φ(˜

q ,φ)

˜

= dFF∗ −1 (˜q) (φ)(dπ

˜ dΦq˜,φ

˜)

Φ(˜

q ,φ)

˜

= dFF∗ −1 (˜q) (φ)d(π

◦ Φ)q˜,φ˜

−1

˜

◦π

˜ )q˜,φ˜

= dFF∗ −1 (˜q) (φ)d(F

(2)

−1

˜

= φ(dF

˜)

F −1 (˜

q ) dFq˜ dπq˜,φ

˜

= φ(dπ

˜)

q˜,φ

˜

= λQ

˜ .

(˜

q ,φ)

To go from the third equality to the fourth one, recall that we showed that Φ is a bundle homomorphism between the cotangent bundles (cf proposition 3.1), i.e. F −1 ◦ π

˜ = π ◦ Φ. This proves

˜

∗ Q

Q

that Φ λ = λ . Furthermore, the property of the pullback ensures that

˜

Φ∗ (dλQ ) = d(Φ∗ λQ ),

hence

˜

Φ∗ ω Q = ω Q .

This proves Theorem 4.7 and the corollary follows directly. Notice that it also would have worked

to consider local diffeomorphisms instead of global diffeomorphisms by modifying our functor. If

we define T ∗ to act on local diffeomorphism as follow:

T ∗ f (x, α) = (f (x), α(df (x)−1 ))

where α ∈ Tx∗ Q and x ∈ Q. This yields, by very similar computation, a covariant functor between

the category of smooth manifolds equipped with local diffeomorphisms ManLoc to symplectic

manifolds Symp equipped with symplectomorphisms.

References

[1] John M. Lee, Introduction to Smooth Maniolds. Second edition, Springer, 2012 .

[2] Gil R. Cavalcanti, Differential Geometry Lecture Notes. Utrecht University, 2018.

[3] Fabien Ziltener, Symplectic Geometry Lecture Notes. Utrecht University, 2017.

[4] Fabian Ziltener, Yael Karshon, Classification of Hamiltonian group actions on exact symplectic

manifolds with proper momentum maps 2018

[5] V. S. Varadarajan, Vector bundles and connections in physics and mathematics, some historical

remarks.

[6] Dusa McDuff, Dietmar Salamon, Introduction to Symplectic Topology, Oxford Graduate text in

Mathematics, Third edition, 2016.

13

[7] M. Abouzaid Framed Bordism and Lagrangian Embeddings of Exotic Spheres

[8] T. Ekholm, T. Kragh, I. Smith Lagrangian Exotic Spheres

[9] Wikipedia page on Cotangent Bundles

14

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